3.2697 \(\int (\frac{6 a x^2}{b (4+m) \sqrt{a+b x^{-2+m}}}+\frac{x^m}{\sqrt{a+b x^{-2+m}}}) \, dx\)

Optimal. Leaf size=26 \[ \frac{2 x^3 \sqrt{a+b x^{m-2}}}{b (m+4)} \]

[Out]

(2*x^3*Sqrt[a + b*x^(-2 + m)])/(b*(4 + m))

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Rubi [C]  time = 0.0855866, antiderivative size = 160, normalized size of antiderivative = 6.15, number of steps used = 5, number of rules used = 2, integrand size = 45, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.044, Rules used = {365, 364} \[ \frac{x^{m+1} \sqrt{\frac{b x^{m-2}}{a}+1} \, _2F_1\left (\frac{1}{2},-\frac{m+1}{2-m};\frac{1-2 m}{2-m};-\frac{b x^{m-2}}{a}\right )}{(m+1) \sqrt{a+b x^{m-2}}}+\frac{2 a x^3 \sqrt{\frac{b x^{m-2}}{a}+1} \, _2F_1\left (\frac{1}{2},-\frac{3}{2-m};-\frac{m+1}{2-m};-\frac{b x^{m-2}}{a}\right )}{b (m+4) \sqrt{a+b x^{m-2}}} \]

Antiderivative was successfully verified.

[In]

Int[(6*a*x^2)/(b*(4 + m)*Sqrt[a + b*x^(-2 + m)]) + x^m/Sqrt[a + b*x^(-2 + m)],x]

[Out]

(2*a*x^3*Sqrt[1 + (b*x^(-2 + m))/a]*Hypergeometric2F1[1/2, -3/(2 - m), -((1 + m)/(2 - m)), -((b*x^(-2 + m))/a)
])/(b*(4 + m)*Sqrt[a + b*x^(-2 + m)]) + (x^(1 + m)*Sqrt[1 + (b*x^(-2 + m))/a]*Hypergeometric2F1[1/2, -((1 + m)
/(2 - m)), (1 - 2*m)/(2 - m), -((b*x^(-2 + m))/a)])/((1 + m)*Sqrt[a + b*x^(-2 + m)])

Rule 365

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[(a^IntPart[p]*(a + b*x^n)^FracPart[p])
/(1 + (b*x^n)/a)^FracPart[p], Int[(c*x)^m*(1 + (b*x^n)/a)^p, x], x] /; FreeQ[{a, b, c, m, n, p}, x] &&  !IGtQ[
p, 0] &&  !(ILtQ[p, 0] || GtQ[a, 0])

Rule 364

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(a^p*(c*x)^(m + 1)*Hypergeometric2F1[-
p, (m + 1)/n, (m + 1)/n + 1, -((b*x^n)/a)])/(c*(m + 1)), x] /; FreeQ[{a, b, c, m, n, p}, x] &&  !IGtQ[p, 0] &&
(ILtQ[p, 0] || GtQ[a, 0])

Rubi steps

\begin{align*} \int \left (\frac{6 a x^2}{b (4+m) \sqrt{a+b x^{-2+m}}}+\frac{x^m}{\sqrt{a+b x^{-2+m}}}\right ) \, dx &=\frac{(6 a) \int \frac{x^2}{\sqrt{a+b x^{-2+m}}} \, dx}{b (4+m)}+\int \frac{x^m}{\sqrt{a+b x^{-2+m}}} \, dx\\ &=\frac{\sqrt{1+\frac{b x^{-2+m}}{a}} \int \frac{x^m}{\sqrt{1+\frac{b x^{-2+m}}{a}}} \, dx}{\sqrt{a+b x^{-2+m}}}+\frac{\left (6 a \sqrt{1+\frac{b x^{-2+m}}{a}}\right ) \int \frac{x^2}{\sqrt{1+\frac{b x^{-2+m}}{a}}} \, dx}{b (4+m) \sqrt{a+b x^{-2+m}}}\\ &=\frac{2 a x^3 \sqrt{1+\frac{b x^{-2+m}}{a}} \, _2F_1\left (\frac{1}{2},-\frac{3}{2-m};-\frac{1+m}{2-m};-\frac{b x^{-2+m}}{a}\right )}{b (4+m) \sqrt{a+b x^{-2+m}}}+\frac{x^{1+m} \sqrt{1+\frac{b x^{-2+m}}{a}} \, _2F_1\left (\frac{1}{2},-\frac{1+m}{2-m};\frac{1-2 m}{2-m};-\frac{b x^{-2+m}}{a}\right )}{(1+m) \sqrt{a+b x^{-2+m}}}\\ \end{align*}

Mathematica [A]  time = 0.170092, size = 26, normalized size = 1. \[ \frac{2 x^3 \sqrt{a+b x^{m-2}}}{b (m+4)} \]

Antiderivative was successfully verified.

[In]

Integrate[(6*a*x^2)/(b*(4 + m)*Sqrt[a + b*x^(-2 + m)]) + x^m/Sqrt[a + b*x^(-2 + m)],x]

[Out]

(2*x^3*Sqrt[a + b*x^(-2 + m)])/(b*(4 + m))

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Maple [A]  time = 0.029, size = 40, normalized size = 1.5 \begin{align*} 2\,{\frac{ \left ( a{x}^{2}+b{x}^{m} \right ) x}{ \left ( 4+m \right ) b}{\frac{1}{\sqrt{{\frac{a{x}^{2}+b{x}^{m}}{{x}^{2}}}}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(6*a*x^2/b/(4+m)/(a+b*x^(-2+m))^(1/2)+x^m/(a+b*x^(-2+m))^(1/2),x)

[Out]

2*x*(a*x^2+b*x^m)/b/(4+m)/((a*x^2+b*x^m)/x^2)^(1/2)

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Maxima [A]  time = 1.17936, size = 50, normalized size = 1.92 \begin{align*} \frac{2 \,{\left (a x^{4} + b x^{2} x^{m}\right )}}{\sqrt{a x^{2} + b x^{m}} b{\left (m + 4\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(6*a*x^2/b/(4+m)/(a+b*x^(-2+m))^(1/2)+x^m/(a+b*x^(-2+m))^(1/2),x, algorithm="maxima")

[Out]

2*(a*x^4 + b*x^2*x^m)/(sqrt(a*x^2 + b*x^m)*b*(m + 4))

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Fricas [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: UnboundLocalError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(6*a*x^2/b/(4+m)/(a+b*x^(-2+m))^(1/2)+x^m/(a+b*x^(-2+m))^(1/2),x, algorithm="fricas")

[Out]

Exception raised: UnboundLocalError

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Sympy [C]  time = 153.193, size = 170, normalized size = 6.54 \begin{align*} \frac{6 a x^{3} \Gamma \left (\frac{3}{m - 2}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{1}{2}, \frac{3}{m - 2} \\ 1 + \frac{3}{m - 2} \end{matrix}\middle |{\frac{b x^{m} e^{i \pi }}{a x^{2}}} \right )}}{b \left (m + 4\right ) \left (\sqrt{a} m \Gamma \left (1 + \frac{3}{m - 2}\right ) - 2 \sqrt{a} \Gamma \left (1 + \frac{3}{m - 2}\right )\right )} + \frac{x x^{m} \Gamma \left (\frac{m}{m - 2} + \frac{1}{m - 2}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{1}{2}, \frac{m}{m - 2} + \frac{1}{m - 2} \\ \frac{m}{m - 2} + 1 + \frac{1}{m - 2} \end{matrix}\middle |{\frac{b x^{m} e^{i \pi }}{a x^{2}}} \right )}}{\sqrt{a} m \Gamma \left (\frac{m}{m - 2} + 1 + \frac{1}{m - 2}\right ) - 2 \sqrt{a} \Gamma \left (\frac{m}{m - 2} + 1 + \frac{1}{m - 2}\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(6*a*x**2/b/(4+m)/(a+b*x**(-2+m))**(1/2)+x**m/(a+b*x**(-2+m))**(1/2),x)

[Out]

6*a*x**3*gamma(3/(m - 2))*hyper((1/2, 3/(m - 2)), (1 + 3/(m - 2),), b*x**m*exp_polar(I*pi)/(a*x**2))/(b*(m + 4
)*(sqrt(a)*m*gamma(1 + 3/(m - 2)) - 2*sqrt(a)*gamma(1 + 3/(m - 2)))) + x*x**m*gamma(m/(m - 2) + 1/(m - 2))*hyp
er((1/2, m/(m - 2) + 1/(m - 2)), (m/(m - 2) + 1 + 1/(m - 2),), b*x**m*exp_polar(I*pi)/(a*x**2))/(sqrt(a)*m*gam
ma(m/(m - 2) + 1 + 1/(m - 2)) - 2*sqrt(a)*gamma(m/(m - 2) + 1 + 1/(m - 2)))

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{m}}{\sqrt{b x^{m - 2} + a}} + \frac{6 \, a x^{2}}{\sqrt{b x^{m - 2} + a} b{\left (m + 4\right )}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(6*a*x^2/b/(4+m)/(a+b*x^(-2+m))^(1/2)+x^m/(a+b*x^(-2+m))^(1/2),x, algorithm="giac")

[Out]

integrate(x^m/sqrt(b*x^(m - 2) + a) + 6*a*x^2/(sqrt(b*x^(m - 2) + a)*b*(m + 4)), x)